Affine Plane Of Order Research Articles

A note on large Kakeya sets

Abstract A Kakeya set 𝓚 in an affine plane of order q is the point set covered by a set 𝓛 of q + 1 pairwise non-parallel lines. By Dover and Mellinger [6], Kakeya sets with size at least q 2 – 3q + 9 contain a large knot, i.e. a point of 𝓚 lying on many lines of 𝓛. We improve on this result by showing that Kakeya set of size at least ≈ q 2 – q + q contain a large knot, and we obtain a sharp result for planes containing a Baer subplane.

A note about monochromatic components in graphs of large minimum degree

For all positive integers $r\geq 3$ and $n$ such that $r^2-r$ divides $n$ and an affine plane of order $r$ exists, we construct an $r$-edge colored graph with minimum degree $(1-\frac{r-2}{r^2-r})n-2$ such that the largest monochromatic component has order less than $\frac{n}{r-1}$. This generalizes an example of Guggiari and Scott and, independently, Rahimi for $r=3$ and thus disproves a conjecture of Gyarfas and Sarkozy for all integers $r\geq 3$ such that an affine plane of order $r$ exists.

Affine and projective planes linked with projective lines over certain rings of lower triangular matrices

Let Tn(q) be the ring of lower triangular matrices of order n≥2 with entries from the finite field F(q) of order q≥2 and let Tn2(q) denote its free left module. For n=2,3 it is shown that the projective line over Tn(q) gives rise to a set of (q+1)(n−1)q3(n−1)(n−2)2 affine planes of order q. The points of such an affine plane are non-free cyclic submodules of Tn2(q) not contained in any non-unimodular free cyclic submodule of Tn2(q) and its lines are points of the projective line. Furthermore, it is demonstrated that each affine plane can be extended to the projective plane of order q, with the ‘line at infinity’ being represented by those free cyclic submodules of Tn2(q) that are generated by non-unimodular pairs. Our approach can be adjusted to address the case of arbitrary n.

A New Lower Bound for the Size of an Affine Blocking Set

A blocking set in an affine plane is a set of points $B$ such that every line contains at least one point of $B$. The best known lower bound for blocking sets in arbitrary (non-desarguesian) affine planes was derived in the 1980's by Bruen and Silverman. In this note, we improve on this result by showing that a blocking set of an affine plane of order $q$, $q\geqslant 25$, contains at least $q+\lfloor\sqrt{q}\rfloor+3$ points.

Blocking and Double Blocking Sets in Finite Planes

In this paper, by using properties of Baer subplanes, we describe the construction of a minimal blocking set in the Hall plane of order $q^2$ of size $q^2+2q+2$ admitting $1-$, $2-$, $3-$, $4-$, $(q+1)-$ and $(q+2)-$secants. As a corollary, we obtain the existence of a minimal blocking set of a non-Desarguesian affine plane of order $q^2$ of size at most $4q^2/3+5q/3$, which is considerably smaller than $2q^2-1$, the Jamison bound for the size of a minimal blocking set in an affine Desarguesian plane of order $q^2$.We also consider particular André planes of order $q$, where $q$ is a power of the prime $p$, and give a construction of a small minimal blocking set which admits a secant line not meeting the blocking set in $1$ mod $p$ points. Furthermore, we elaborate on the connection of this problem with the study of value sets of certain polynomials and with the construction of small double blocking sets in Desarguesian projective planes; in both topics we provide some new results.

All-involution table algebras and finite projective spaces

Table algebras all of whose nonidentity basis elements are involutions (in the sense of Zieschang), which serve as a counterpoint to the generic Hecke algebras parametrized by Coxeter groups, are classified. If two-generated, they are the family Hn (for all n ≥ 3), which for suitable n arise from schemes defined by affine planes of order n − 1. Otherwise, the basis involutions correspond to the points of a finite projective space whose incidence geometry determines the algebra multiplication. This generalizes to table algebras a previous result of van Dam for association schemes. An algebraic characterization is also given.

Embedding Cycles in Finite Planes

We define and study embeddings of cycles in finite affine and projective planes. We show that for all $k$, $3\le k\le q^2$, a $k$-cycle can be embedded in any affine plane of order $q$. We also prove a similar result for finite projective planes: for all $k$, $3\le k\le q^2+q+1$, a $k$-cycle can be embedded in any projective plane of order $q$.

On quasi-thin association schemes

An association scheme is called quasi-thin if each of its basic relations has valency 1 or 2. A quasi-thin scheme is called Kleinian if its thin residue is the Klein four-group with respect to relational multiplication. It is proved that any Kleinian quasi-thin scheme arises from a near-pencil on 3 points, from an affine plane of order 2, or from a projective plane of order 2. The main result in this paper is that any non-Kleinian quasi-thin scheme is schurian and separable. We also construct an infinite family of Kleinian quasi-thin schemes which is neither schurian nor separable.

Steiner triple systems satisfying the 4-vertex condition

Higman asked which block graphs of Steiner triple systems of order v satisfy the 4-vertex condition and left the cases v = 9, 13, 25 unsettled.We give a complete answer to this question by showing that the affine plane of order 3 and the binary projective spaces are the only such systems. The major part of the proof is to show that no block graph of a Steiner triple system of order 25 satisfies the 4-vertex condition.

On some operations on finite affine planes applied to the theory of regular configurations

AbstractWe consider some operations on affine planes which resemble the construction of a derived affine plane at a point of the Benz plane. We call them Benz-contractions (B-contractions), distinguishing between chain contractions and generator contractions. We prove that the Pappos–Pascal configuration is the B-contraction of the affine plane of order 4 and we relate it to the Havliček–Tietze configuration. We present a new (HT)

A construction of 2-designs from Steiner systems and extendable designs

We give a construction of a 2- ( m n 2 + 1 , m n , ( n + 1 ) ( m n − 1 ) ) design starting from a Steiner system S ( 2 , m + 1 , m n 2 + 1 ) and an affine plane of order n. This construction is applied to known classes of Steiner systems arising from affine and projective geometries, Denniston designs, and unitals. We also consider the extendability of these designs to 3-designs.

On connected line sets of antiflag class [formula omitted] in [formula omitted

We investigate the partial linear spaces, fully embedded in an affine space AG ( n , q ) with the property that for every antiflag { p , L } , the number of lines through p intersecting L is either 0 , α , or q . Besides some general results we prove a complete classification of those geometries fully embedded in an affine plane of order q and of the connected geometries with 1 < α < q , fully embedded in AG ( 3 , q ) .

Möbius-Kantor configurations in the affine plane of order 7

We partition the affine plane of order 7 into a set of Mobius-Kantor configurations 83 plus a set consisting only of one point.

Quasi-affine symmetric designs

A symmetric design with parameters v = q 2(q + 2), k = q(q + 1), ? = q, q ? 2, is called a quasi-affine design if its point set can be partitioned into q + 2 subsets P 0, P 1,..., P q , P q+1 such that the induced structure in every point neighborhood is an affine plane of order q (repeated q times). A quasi-affine design with q ? 3 determines its point neighborhoods uniquely and dual of such a design is also a quasi-affine design. These structural properties pave way for definition of a strongly quasi-affine design and it is also shown that associated with every quasi-affine design is a unique strongly quasi-affine design from which the given quasi-affine design is obtained by certain unique cutting and pasting operation. This investigation also enables us to associate a unique 2-regular graph with q + 2 vertices and in turn, a unique colored partition of the integer q + 2. These combinatorial consequences are finally used to obtain an exponential lower bound on the number of non-isomorphic solutions of such symmetric designs improving the earlier lower bound of 2.

A thin near hexagon with 50 points

We show the existence and uniqueness of a thin near hexagon which has 50 points and an affine plane of order 3 as a local space.

A Theorem Concerning Nets Arising from Generalized Quadrangles with a Regular Point

Suppose \mathcal{S} is a generalized quadrangle (GQ) of order (s, t), s, t \ne 1, with a regular point. Then there is a net which arises from this regular point. We prove that if such a net has a proper subnet with the same degree as the net, then it must be an affine plane of order t. Also, this affine plane induces a proper subquadrangle of order t containing the regular point, and we necessarily have that s = t^2. This result has many applications, of which we give one example. Suppose \mathcal{S} is an elation generalized quadrangle (EGQ) of order (s, t), s, t \ne 1, with elation point p. Then \mathcal{S} is called a skew translation generalized quadrangle (STGQ) with base-point p if there is a full group of symmetries about p of order t which is contained in the elation group. We show that a GQ \mathcal{S} of order s is an STGQ with base-point p if and only if p is an elation point which is regular.

Complete Sets of Mutually Orthogonal Hypercubes and Their Connections to Affine Resolvable Designs

Recently, Laywine and Mullen proved several generalizations of Bose's equivalence between the existence of complete sets of mutually orthogonal Latin squares of order n and the existence of affine planes of order n. Laywine further investigated the relationship between sets of orthogonal frequency squares and affine resolvable balanced incomplete block designs. In this paper we generalize several of Laywine's results that were derived for frequency squares. We provide sufficient conditions for construction of an affine resolvable design from a complete set of mutually orthogonal Youden frequency hypercubes; we also show that, starting with a complete set of mutually equiorthogonal frequency hypercubes, an analogous construction can always be done. In addition, we give conditions under which an affine resolvable design can be converted to a complete set of mutually orthogonal Youden frequency hypercubes or a complete set of mutually equiorthogonal frequency hypercubes.

Ternary dual codes of the planes of order nine

We determine the minimum weight of the ternary dual codes of each of the four projective planes of order 9, and of the seven affine planes of order 9. The proof includes a construction of a word of small, sometimes minimal, weight in the dual code of any plane of square order containing a Baer subplane.

Flag-Transitive affine planes of order at most 125

We complete the determination of all the flag-transitive affine planes of order at most 125. In particular, all such planes of order 81 are determined. We show that those with a solvable collineation group are obtained by constructions due to Kantor and Suetake.

The Non-Collinearity Graph of the O^+ (8,2)Quadric Is Uniquely Geometrisable

The graph of the title has the points of the O^+(8,2) polar space as its vertices, two such vertices being adjacent iff the corresponding points are non-collinear in the polar space. We prove that, upto isomorphism, there is a unique partial geometry pg(8,7,4) whose point graph is this graph. This is the partial geometry of Cohen, Haemers and Van Lint and De Clerck, Dye and Thas. Our uniqueness proof shows that this geometry has a subgeometry isomorphic to the affine plane of order three, and the geometry is canonically describeable in terms of this affine plane.